A solution to the problem of useful random signal extraction from a
mixture with a noise in the multiplicative observation model is proposed. Unlike
conventional filtering tasks, in the problem under consideration it is supposed that
the distribution (and the model) of the useful signal is unknown. Therefore, in
this case one cannot apply such well-known techniques like Kalman filter or posterior
Stratonovich-Kushner evolution equation. The new paper is a continuation
and development of the author’s article, reported at the First ISNPS Conference
(Halkidiki’2012), where the filtering problem of positive signal with the unknown
distribution had been solved using the generalized filtering equation and nonparametric
kernel techniques. In the present study, new findings are added concerning
the construction of stable procedures for filtering, the search for optimal smoothing
parameter in the multidimensional case and some of the convergence results of the
proposed techniques. The main feature of the problem is the positive distribution support.
In this case, the classical methods of nonparametric estimation with symmetric
kernels are not applicable because of large estimator bias at the support boundary.
To overcome this drawback, we use asymmetric gamma kernel functions. To have
stable estimators, we propose a regularization procedure with a data-driven optimal
regularization parameter. Similar filtering algorithms can be used, for instance, in
the problems of volatility estimation in statistical models of financial and actuarial
mathematics.